Senator Patrick Leahy: 'This has reached the point where judges are being voted on for political reasons, not qualifications.' Photograph: Win Mcnamee/Getty Images

Democrats threatened a major constitutional showdown over Senate filibuster rules on Tuesday after Republicans blocked a third successive woman nominated to sit on a federal appeals court.

Nina Pillard, a Georgetown law professor, would have been only the sixth woman in 120 years to serve on the US court of appeals for the District of Columbia circuit, which is known as the second highest court in the land due to its key role in ruling on legal challenges against the US government.

However, Senate Republicans voted to block passage of legislation confirming her nomination, using rules that require a 60-vote majority to prevent a filibuster, or indefinite procedural delay.

Angry Democrats decried the move – the third involving a woman nominated to the court by president Barack Obama – which they said broke an informal agreement between the parties that judicial confirmations should require only 51 of the 100 senators to vote in favour.

"I have seen more filibusters in a year than I have seen in 35 years. I think we are at the point where there will have to be a rules change," said senator Patrick Leahy, Democratic chairman of the judiciary committee. "This has reached the point where judges are being voted on for political reasons, not qualifications. You do that and you are going to destroy the credibility of the federal courts."

It is unclear how such a rule change might be accomplished, particularly as past attempts have been scuppered by some Democrats anxious to avoid giving up their own power to block legislation were they to lose their current numerical advantage in the Senate.

But a series of apparently gender-driven attacks against the judicial nominees on the basis of their previous opinions on abortion and women's rights has angered much of the party, leading to shouting on the floor of the Senate from Leahy, who said the time had come for "drastic action".

Democratic whip Dick Durbin said: "We have been here before, but there comes a tipping point and I am afraid we have reached it."

Senator Amy Klobuchar said: "The so-called gentleman's agreement has not treated these women very well."

Republicans dismissed claims that their opposition was driven by the gender of the nominees, arguing instead that Obama was unnecessarily "stacking" the court with ideological appointees to help him circumvent Congress.

"I have heard the other side say that Republicans are opposing the nominee because of her gender," said senator Chuck Grassley. "That is grossly offensive. It is a well-worn card played against us."

Grassley claimed the White House was planning to overcome recent gridlock in Congress by issuing so-called executive orders on issues such as climate change and gun control and needed a Democratic majority on the court to avoid legal challenge.

"The only way the president can successfully bypass Congress is if he stacks the court with ideological allies," he said.

Republican whip John Cornyn claimed this justified ignoring previous agreements to hold simple majority votes on nominations. "The desire to make the court a liberal rubber stamp creates the extraordinary circumstance that justifies a filibuster," he said.

The GOP sought to argue that the court, which has four judges nominated from each party but a majority of Republican semi-retired justices, has sufficient staff to handle its workload.

But it is the party's criticism of nominees such as Pillard for their past comments on fertility issues that have sparked most controversy.

"Reproductive rights, including the rights to contraception and abortion, play a central role in freeing women from historically routine conscription into maternity," wrote Pillard. "It is reproductive rights that have begun to allow women to decide whether and when to follow the path of motherhood."

On Tuesday, Durbin said it was "shameful" that Republicans had criticised her nomination because of such statements. "It is no secret she's written articles arguing for equality between men and women. Some find this radical thinking; most American people think it should be the law of the land," he said.

He attacked the "old trick" of frustrating nominations using a filibuster: "To do it to an amendment is bad enough, but to do it to a human being is something we should think long and hard about."